# Michael Sipser Edition 3 Exercise 2 Question 8 (Page No. 155)

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Show that the string the girl touches the boy with the flower has two different leftmost derivations in grammar $G_2$ on $\text{page 103.}$ Describe in English the two different meanings of this sentence.

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Give context-free grammars generating the following languages. The set of strings over the alphabet $\{a,b\}$ with more $a's$ than $b's$ The complement of the language $\{a^{n}b^{n}\mid n\geq 0\}$ $\{w\#x\mid w^{R}$ $\text{is a substring of$x$for }$ $w,x \in\{0,1\}^{*}\}$ ... $\text{each}$ $x_{i}\in\{a,b\}^{*},$ $\text{and for some i and }$ $j,x_{i}=x_{j}^{R}\}$
Give context-free grammars that generate the following languages$.$ In all parts, the alphabet $\sum$ is $\{0,1\}.$ $\text{{w| w contains at least three 1’s}}$ $\text{{w| w starts and ends with the same symbol}}$ $\text{{w| the length of w is odd}}$ $\text{{w| the length of w is odd and its middle symbol is a 0}}$ $\text{{$w|w=w^{R},$that is, w is a palindrome}}$ $\text{The empty set}.$
Let $\Sigma = \{a,b\}.$ Give a $CFG$ generating the language of strings with twice as many $a’s$ as $b’s.$ Prove that your grammar is correct$.$
Let $CFG$ $G$ be the following grammar$.$ $S\rightarrow aSb \mid bY \mid Y a$ $Y\rightarrow bY \mid aY \mid \epsilon$ Give a simple description of $L(G)$ in English$.$ Use that description to give a $CFG$ for $\overline{L(G)},$ the complement of $L(G).$